How to convert an edge into a curved edge?
I think we lack informations about what's the final goal, for example are you supposed to extrude, smooth, etc?

But if you want a simple flat shape, you can round the corners this way:


Delete the faces with a X > Only Faces.
Select the vertices that are supposed to be round.
Press shiftctrlB to bevel. Press + to increase the amount of segments, or do it on the bottom of the Tools panel (on the left of your 3D view).
Now fill the whole mesh with an F.
Finish the mesh with some additional vertices and edges if necessary, the knife tool is convenient for this purpose.